Average Density of States for Hermitian Wigner Matrices

TL;DR

This paper proves that for large Hermitian Wigner matrices with regular entry distributions, the expected density of states converges to the semicircle law on very small intervals, confirming a key aspect of random matrix theory.

Contribution

It establishes the convergence of the expected density of states to the semicircle law on arbitrarily small intervals for Hermitian Wigner matrices with regular distributions.

Findings

Expected density of states converges to semicircle law

Convergence holds on arbitrarily small intervals

Results apply to large N asymptotics

Abstract

We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on {\it arbitrarily} small intervals converges to the semicircle law, as $N$ tends to infinity.